Frechet derivatives in inverse obstacle scattering

Abstract

There is an error in theorem 2.1 of this paper, which shows a representation of the Fréchet derivative with respect to the boundary of the far field operator from scattering by obstacles with Robin boundary conditions. The boundary condition in equation (2.4) has to be replaced by where H denotes the mean curvature of the boundary. The error occurs in the derivation of equation (2.12). Therefore the proof of theorem 2.1 presented in this paper has to be modified on pages 376-7 along the following lines. For the surface integral we obtain This can be seen by considering a fixed point and a local parametrization , of with The perturbed boundary is then locally described by . We compute at the point U0 with and the orthonormality of the tangential vectors leads to We obtain This holds independently of the parametrization and uniformly on . With the estimates (2.10)-(2.12) we first see from (2.8) by a perturbation argument that tends to u in . This is well known: the scattered field depends continuously on the boundary, or in our notation FR is a continuous operator (cf Kress and Zinn (1992) for the Dirichlet problem). Therefore, with equation (2.8) the proof is completed by showing for all v. Since u' is a radiating solution of (2.3)-(2.4) we obtain from the boundary condition Now we proceed as in the paper with the only modification of adding the integral in any following computation of the sesquilinear form . This rectifies the proof. Reference Kress R and Zinn A 1992 On the numerical solution of the three-dimensional inverse obstacle scattering problem J. Comput. Appl. Math. 42 46-61